Visibility of Mordell-Weil groups.
has connected fibers.
Computation of -adic heights and log convergence.
Visibility of the Shafarevich-Tate group at higher level.
On the generation of the coefficient field of a newform by a single Hecke eigenvalue
Let be a non-CM newform of weight . Let be a subfield of the coefficient field of . We completely settle the question of the density of the set of primes such that the -th coefficient of generates the field . This density is determined by the inner twists of . As a particular case, we obtain that in the absence of nontrivial inner twists, the density is for equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions...
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